A simple group is pseudofinite if and only if it is isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that any simple pseudofinite group $G$ is finite-dimensional. In particular, if $\operatorname{dim}(G)=3$ then $G$ is isomorphic to $\operatorname{PSL}(2, F)$ for some pseudofinite field $F$. In this talk, we describe the structures of finite-dimensional pseudofinite groups with dimension $<4$, without using CFSG. In the case $\operatorname{dim}(G)=3$ we show that either $G$ is soluble-by-finite or has a finite normal subgroup $Z$ so that $G / Z$ is a finite extension of $\operatorname{PSL}(2, F)$. This in particular implies that the classification $G \cong \operatorname{PSL}(2, F)$ from the above does not require CFSG. This is joint work with Frank Wagner.[-]

Roth's theorem states that a subset $A$ of $\{1, \ldots, N\}$ of positive density contains a positive $N^2$-proportion of (non-trivial) three arithmetic progressions, given by pairs $(a, d)$ with $d \neq 0$ such that $a, a+d, a+2 d$ all lie in $A$. In recent breakthrough work by Kelley and Meka, the known bounds have been improved drastically. One of the core ingredients of the their proof is a version of the almost periodicity result due to Croot and Sisask. The latter has been obtained in a non-quantitative form by Conant and Pillay for amenable groups using continuous logic.
In joint work with Daniel Palacín, we will present a model-theoretic version (in classical first-order logic) of the almost-periodicity result for a general group equipped with a Keisler measure under some mild assumptions and show how to use this result to obtain a non-quantitative proof of Roth's result. One of the main ideas of the proof is an adaptation of a result of Pillay, Scanlon and Wagner on the behaviour of generic types in a definable group in a simple theory.[-]

We give an arithmetic version of Tao's algebraic regularity lemma (which was itself an improved Szemerédi regularity lemma for graphs uniformly definable in finite fields). In the arithmetic regime the objects of study are pairs $(G, D)$ where $G$ is a group and $D$ an arbitrary subset, all uniformly definable in finite fields. We obtain optimal results, namely that the algebraic regularity lemma holds for the associated bipartite graph $(G, G, E)$ where $E(x, y)$ is $x y^{-1} \in D$, witnessed by a the decomposition of $G$ into cosets of a uniformly definable small index normal subgroup $H$ of $G$.[-]

A primitive permutation group $(X, G)$ is a group $G$ together with an action of $G$ on $X$ such that there are no nontrivial equivalence relations on $X$ preserved by $G$. An rough classification of primitive permutation groups of finite Morley rank, modeled on the O'Nan-Scott theorem for finite primitive permutation groups, has been carried out by Macpherson and Pillay and this classification was then used by Borovik and Cherlin to prove that if $(X, G)$ is a primitive permutation group of finite Morley rank, the rank of $G$ can be bounded in terms of the rank of $X$. We study the analogous situation for pseudo-finite primitive permutation groups of finite SU-rank, building both on supersimple group theory and classification results of Liebeck-Macpherson-Tent. This is joint work in progress with Ulla Karhumäki.[-]

My main goal is to present a somewhat technical result, joint with Karim Adiprasito and Ehud Hrushovski, about the existence of a family of polynomials whose zeros (at a certain desired family of sets) increase faster than their degree.I will explain the 'local' and 'global' consequences of this result. In the unlikely case I have enough time, I shall attempt to elaborate on the relation to quantifier-free stability in Globally Valued Fields.This will be in some sense complementary to my recent talk in Banff.[-]